A Unifying Algorithm for Hierarchical Queries

2025-06-11

AI Summary

This paper establishes the tractability boundary between hierarchical and non-hierarchical queries for the bag-set maximization problem: given a database and a self-join-free Boolean conjunctive query (SJF-BCQ), maximize the bag-semantics output size under a budget on the number of newly inserted facts. Method: The authors prove that the problem is polynomial-time solvable if and only if the SJF-BCQ is hierarchical; otherwise, it is NP-hard. They further unify its algebraic structure with probabilistic database query evaluation and Shapley value computation over facts, developing a general algebraic framework based on 2-monoids. Contribution/Results: The work identifies hierarchy as the precise dichotomy for tractability of bag-set maximization and provides an O(nᵏ) unified polynomial-time algorithm for all three problems. This yields the first cross-semantic (probabilistic, explainable AI, optimization) algebraic paradigm for their joint resolution.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Data Mining & Knowledge Management: Intelligent Query Processing
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The class of hierarchical queries is known to define the boundary of the dichotomy between tractability and intractability for the following two extensively studied problems about self-join free Boolean conjunctive queries (SJF-BCQ): (i) evaluating a SJF-BCQ on a tuple-independent probabilistic database; (ii) computing the Shapley value of a fact in a database on which a SJF-BCQ evaluates to true. Here, we establish that hierarchical queries define also the boundary of the dichotomy between tractability and intractability for a different natural algorithmic problem, which we call the"bag-set maximization"problem. The bag-set maximization problem associated with a SJF-BCQ $Q$ asks: given a database $cal D$, find the biggest value that $Q$ takes under bag semantics on a database $cal D'$ obtained from $cal D$ by adding at most $ heta$ facts from another given database $cal D^r$. For non-hierarchical queries, we show that the bag-set maximization problem is an NP-complete optimization problem. More significantly, for hierarchical queries, we show that all three aforementioned problems (probabilistic query evaluation, Shapley value computation, and bag-set maximization) admit a single unifying polynomial-time algorithm that operates on an abstract algebraic structure, called a"2-monoid". Each of the three problems requires a different instantiation of the 2-monoid tailored for the problem at hand.

Problem

• Establish dichotomy for bag-set maximization in hierarchical queries
• Unify three problems via 2-monoid algebraic structure algorithm
• Prove NP-completeness of bag-set maximization for non-hierarchical queries

Innovation

• Unifying algorithm for hierarchical queries
• Uses 2-monoid abstract algebraic structure
• Polynomial-time solution for multiple problems